F. Lemieux, Finite Groupoids and their Applications to Computational Complexity. Ph. D. Thesis, School of Computer Science, McGill University, Montreal (1996).  Home/Search   Document Details and Download   Summary   Related Articles   Check

....n . For example we must be able to determine if a segment of the program is the left child of another segment, etc. In this paper, it will be sufficient to use logspace uniformity, and so we say that a family of programs is uniform if it is L uniform. For more details concerning the uniformity see [7]. A parenthesized program is said to be left to right if the evaluation tree is such that the right child of any inner node is a leaf. An evaluation tree is called linear is for any inner node, at most one child is not a leaf. A linear program (deterministic or not) is one where parenthesizations ....

....is in SAC 1 if and only if it is recognized by uniform programs over a constant order (polynomial order) groupoid. b) 8] A language is in NP if and only if it is recognized by uniform programs over exponential order groupoids. Without loss of generality, the programs can be linear (see [7]) c) 7] A language is in NL if and only if it is recognized by uniform linear programs over constant order (polynomial order) groupoids. d) 7] A language is in NC 1 if and only if it is recognized by uniform parenthesized programs over constant order groupoids. 3 Programs and Turing ....

[Article contains additional citation context not shown here]

F. Lemieux, Finite groupoids and their applications to computational complexity, Ph.D. Thesis, McGill University, May 1996.

....level by level. An algebraic circuit over an algebra (A; Delta) is a circuit where each gate outputs the product a Delta b of its inputs, rather than implementing the standard Boolean functions; the Circuit Value problem for various classes of algebras has been studied by a number of authors [3, 4, 27]. Predicting an r = 1=2 CA is clearly a special case of Circuit Value, where the circuit has a simple periodic structure in space and time. However, a number of the results we prove below for CAs will in fact hold for algebraic circuits of arbitrary shape. As shorthand, we will say a CA is in NC ....

F. Lemieux, Finite Groupoids and their Applications to Computational Complexity. Ph. D. Thesis, School of Computer Science, McGill University, Montr'eal (1996).

....at the output, MODp gates on the middle level, and small fan in AND gates at the inputs. 5.3 Solvability versus Nonsolvability Another body of lower bounds comes directly from the algebraic characterization of ACC 0 . For more background about this approach to circuit complexity, see [MPT91, Lem96] Recall that few lower bounds are known even for circuits consisting only of MOD6 gates. A natural conjecture is that these circuits cannot compute the AND function (just as AC 0 circuits cannot compute MOD6) The first lower bounds in this direction appear in [BST90] where the authors show ....

F. Lemieux. Finite Groupoids and their Applications to Computational Complexity. PhD thesis, McGill University, 1996.

....L a and R a are the identity 1. Then aL a (b) L a (b) b, so anb = L a (b) a(a( a z n 1 times b) is in P(Q) Similarly for a=b; then by composition we can de ne [a; b] ab = ba, a; b; c] ab)c = a(bc) a = 1=a and a = an1. Then we have the following [6, 11]: Theorem 3.5 If G is a nite simple loop that is not an Abelian group then G is functionally complete. Proof. Let g 1 ; g 2 2 G and g 1 6= 1. By Lemma 3.3, there exists a function g1 g 2 (x) that xes the identity and maps g 1 to g 2 and that is expressible in G. Because it is simple and ....

F. Lemieux, Finite Groupoids and their Applications to Computational Complexity. Ph. D. Thesis, School of Computer Science, McGill University, Montreal (1996).

....group has a simple non Abelian divisor, non solvable groups are Boolean complete by lemma 1. ut (In fact, 2] seems to state that non solvable groups are strongly Booleancomplete; we believe this is a slight mistake. By using an associator instead of a commutator, this generalizes to loops [13]. Like the commutator, the associator [x; y; z] is 1 if any of its arguments is 1, since the identity associates with everything. Theorem 6. Simple non Abelian loops are strongly Boolean complete, and nonsolvable loops are Boolean complete. Proof. Assume without loss of generality that G is ....

....a b = x;y;z] t [ t x (a) t y (b) z] and :a = t=a or t Delta a . So simple non Abelian loops are strongly Booleancomplete; and since non solvable loops have simple non Abelian divisors, they are Boolean complete by lemma 1. ut In fact, simple non Abelian groups [15] simple loops [13], and non affine simple quasigroups [16] have a stronger property, that their polynomial closure contains all possible n ary functions on their elements. This is called functional completeness, and is of interest in the field of multi valued logic [22, 25] However, Boolean completeness is ....

F. Lemieux, Finite Groupoids and their Applications to Computational Complexity. Ph. D. Thesis, School of Computer Science, McGill University, Montr'eal (1996).

....separations between them. Most of this work has dealt with associative structures, namely groups, semigroups and monoids, largely because the idea of a syntactic monoid is familiar from the theory of nite state automata. However, some progress has been made in the non associative case as well [5, 6, 13, 10, 17]. Here, concepts such as solvability generalize in several competing ways, and nding the appropriate one for a given problem can be dicult. For instance, the complexity of circuit evaluation and expression evaluation over loops is determined by two di erent generalizations of solvability, which ....

F. Lemieux, Finite Groupoids and their Applications to Computational Complexity. Ph. D. Thesis, School of Computer Science, McGill University, Montreal (1996).

....[x;y] t Gamma [ t x (a) t y (b) Delta This expression evaluates to t if a = b = t, and 1 if either a or b is 1; in other words, it is an and gate. Finally, we can express negation as :a = t Delta a Gamma1 . ut By using an associator instead of a commutator, this generalizes to loops [12]. Like the commutator, the associator [x; y; z] is 1 if any of its arguments is 1, since the identity associates with everything. Theorem 5. Non solvable loops are strongly Boolean complete. Proof. Assume without loss of generality that G is simple and non associative, since theorem 4 treats the ....

....associative case. Choose a triplet of non associating elements x; y; z 2 G with [x; y; z] 6= 1. Let false = 1 and true = t 6= 1 as before. Then a b = x;y;z] t Gamma [ t x (a) t y (b) z] Delta and :a = t=a or t Delta a ae . ut In fact, simple non Abelian groups [14] simple loops [12], and non affine simple quasigroups [15] have a stronger property, that their polynomial closure contains all possible n ary functions on their elements. This is called functional completeness, and is of interest in the field of multi valued logic [21, 24] However, Boolean completeness is ....

F. Lemieux, Finite Groupoids and their Applications to Computational Complexity. Ph. D. Thesis, School of Computer Science, McGill University, Montr'eal (1996).

....a rich theory with many deep results and applications, and it remains an active field that continues to challenge researchers. This makes more striking the observation that no such theory exists for context free languages. Nevertheless, this topic has been the subjet of recent investigations (e.g. [18, 21, 10, 13, 19, 7, 8, 20]) that we briefly describe here. A groupoid G is a set with a binary operation that can be non associative. All groupoids considered in this paper are finite. Groupoids can be used as language recognizers as follows. For any w 2 G , denote with G(w) the set of all elements g 2 G such that w can ....

F. Lemieux, Finite Groupoids and their Applications to Computational Complexity, Ph.D. Thesis, McGill University, May 1996.

....be a group such as SL(2; 5) whose simple divisor PSL(2; 5) A 5 is not a subgroup [33] recall that a divisor is not generally a subgroup) In this case, PSL(2; 5) is strongly Boolean complete, while SL(2; 5) is not. By using an associator instead of a commutator, this generalizes to loops [18, 11]. Like the commutator, the associator [x; y; z] is 1 if any of its arguments is 1, since the identity associates with everything. Theorem 6. Simple non Abelian loops are strongly Boolean complete, and nonsolvable loops are Boolean complete. Proof. Assume without loss of generality that G is ....

....these are not the same for all a unless the loop has the right inverse property) So simple non Abelian loops are strongly Boolean complete; and since non solvable loops have simple nonAbelian divisors, they are Boolean complete by Lemma 1. ut In fact, simple non Abelian groups [20] simple loops [18, 11], and non ane simple quasigroups [21] have a stronger property, that their closure contains all possible n ary functions on their elements. This is called functional completeness, and is of interest in the eld of multi valued logic [27, 31] However, Booleancompleteness is sucient for our ....

F. Lemieux, Finite Groupoids and their Applications to Computational Complexity. Ph. D. Thesis, School of Computer Science, McGill University, Montreal (1996).

....a are the identity 1. Then aL n 1 a (b) L n a (b) b, so anb = L n 1 a (b) a(a( a z n 1 times b) is in P(Q) Similarly for a=b; then by composition we can de ne [a; b] ab = ba, a; b; c] ab)c = a(bc) a = 1=a and a = an1. Then we have the following [6, 11]: Theorem 3.5 If G is a nite simple loop that is not an Abelian group then G is functionally complete. Proof. Let g 1 ; g 2 2 G and g 1 6= 1. By Lemma 3.3, there exists a function g1 g 2 (x) that xes the identity and maps g 1 to g 2 and that is expressible in G. Because it is simple and ....

F. Lemieux, Finite Groupoids and their Applications to Computational Complexity. Ph. D. Thesis, School of Computer Science, McGill University, Montreal (1996).

No context found.

F. Lemieux, Finite groupoids and their applications to computational complexity, Ph.D. Thesis, McGill University, May 1996.

....two cases when the operation is associative. In the non associative setting, the WordProblem has been taken to mean the following: given an unbracketed sequence of elements of A, does there exist a parenthesization that evaluates to some target element; this question has been extensively studied [17, 7, 5] and will not be pursued here. Our goal is to investigate how the algebraic properties of (A; Delta) such as associativity, commutativity, solvability and so on, affect the computational complexity of the problems Expression Evaluation and Circuit Value. Our work is the direct continuation of ....

....all a 2 A) the variables x i (for all 1 i k) and the products (OE 1 Delta OE 2 ) for all OE 1 ; OE 2 2 P(A) We will refer to the set of functions on k variables as P k (A) for instance, P 1 (A) contains the multiplication semigroup M(A) as well as functions like OE(x) x 2 . In [17] it is proved that if G is a simple non abelian loop then P n (G) contains all functions f : G n G. This property is called functional completeness. Not every groupoid is functionally complete, however. For example, Straubing has shown [23] that if G is a solvable group and if false and ....

[Article contains additional citation context not shown here]

F. Lemieux, Finite Groupoids and their Applications to Computational Complexity. Ph. D. Thesis, School of Computer Science, McGill University, Montr'eal (1996).

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC